Integral Representation Research Articles

Numerical Approximation of Gaussian Random Fields on Closed Surfaces

Abstract We consider the numerical approximation of Gaussian random fields on closed surfaces defined as the solution to a fractional stochastic partial differential equation (SPDE) with additive white noise. The SPDE involves two parameters controlling the smoothness and the correlation length of the Gaussian random field. The proposed numerical method relies on the Balakrishnan integral representation of the solution and does not require the approximation of eigenpairs. Rather, it consists of a sinc quadrature coupled with a standard surface finite element method. We provide a complete error analysis of the method and illustrate its performances in several numerical experiments.

On orthogonal Laurent polynomials related to the partial sums of power series

Let \(f(z) = \sum_{k=0}^\infty d_k z^k\), \(d_k\in\mathbb{C}\backslash\{ 0 \}\), \(d_0=1\), be a power series with a non-zero radius of convergence \(\rho\): \(0 <\rho \leq +\infty\). Denote by \(f_n(z)\) the \(n\)-th partial sum of \(f\), and \(R_{2n}(z) = \frac{ f_{2n}(z) }{ z^n }\), \(R_{2n+1}(z) = \frac{ f_{2n+1}(z) }{ z^{n+1} }\), \(n=0,1,2,\dots\). By the general result of Hendriksen and Van Rossum there exists a unique linear functional \(\mathbf{L}\) on Laurent polynomials, such that \(\mathbf{L}(R_n R_m) = 0\), when \(n\not= m\), while \(\mathbf{L}(R_n^2)\not= 0\), and \(\mathbf{L}(1)=1\). We present an explicit integral representation for \(\mathbf{L}\) in the above case of the partial sums. We use methods from the theory of generating functions. The case of finite systems of such Laurent polynomials is studied as well.

Some properties of Ψ-gamma, Ψ-beta and Ψ-hypergeometric matrix functions

Abstract In this paper, we investigate the matrix analogues of the Ψ-beta and Ψ-gamma functions, as well as their properties. With the help of the Ψ-beta matrix function (BMF), we introduce the Ψ-Gauss hypergeometric matrix function (GHMF) and the Ψ-Kummer hypergeometric matrix function (KHMF) and derive certain properties for these matrix functions. Finally, the Ψ-Appell and the Ψ-Lauricella matrix functions are defined by applications of the Ψ-BMF, and their integral representations are also given.

On some identities for confluent hypergeometric functions and Bessel functions

Mathematical functions, which often appear in mathematical analysis, are referred to as special functions and have been studied over hundreds of years. Many books and dictionaries are available that describe their properties and serve as a foundation of current science. In this paper, we find a new integral representation of the Whittaker function of the first kind and show a relevant summation formula for Kummer’s confluent hypergeometric functions. We also perform the specifications of our identities to link to known and new results.

On the Bessel Solution of Kepler’s Equation

Since its introduction in 1650, Kepler’s equation has never ceased to fascinate mathematicians, scientists, and engineers. Over the course of five centuries, a large number of different solution strategies have been devised and implemented. Among them, the one originally proposed by J. L. Lagrange and later by F. W. Bessel still continue to be a source of mathematical treasures. Here, the Bessel solution of the elliptic Kepler equation is explored from a new perspective offered by the theory of the Stieltjes series. In particular, it has been proven that a complex Kapteyn series obtained directly by the Bessel expansion is a Stieltjes series. This mathematical result, to the best of our knowledge, is a new integral representation of the KE solution. Some considerations on possible extensions of our results to more general classes of the Kapteyn series are also presented.

A note on the post quantum-Sheffer polynomial sequences

Abstract In this article, the post quantum analogue of Sheffer polynomial sequences is introduced using concepts of post quantum calculus. The series representation, recurrence relations, determinant expression and certain other properties of this class are established. Further, the 2D-post quantum-Sheffer polynomials are introduced via generating function and their properties are established. Certain identities and integral representations for the 2D-post quantum-Hermite polynomials, 2D-post quantum-Laguerre polynomials, and 2D-post quantum-Bessel polynomials are also considered.

Solvability of nonlinear boundary value problems for non-sloping Timoshenko-type isotropic shells of zero principal curvature

We study the solvability of boundary value problem for a system of second order partial differential equations under boundary given conditions describing the equilibrium of elastic non-sloping isotropic inhomogeneous shells with free boundary in the framework of the Timoshenko shear model. The base of the study method are the integral representations for generalized motions involving arbitrary functions, which also involve arbitrary holomorphic functions. The arbitrary functions ate determined so that the generalized motions satisfy a linear system of equations and linear boundary conditions extracted from the original boundary value problem. The holomorphic functions are sought as Cauchy type integrals with real densities. The integral representations allow us to reduce the initial boundary value problem to a nonlinear operator equations for generalized motions in the Sobolev space. While studying the solvability of this operator equation, the most essential point is to invert it with respect to the linear part. As a result, the work is reduced to an equation, the solvability of which is established on the base of the contracting mapping principle.

M-Srivastava hypergeometric functions: integral representations and solution of fractional differential equations

In this paper, we introduce a new extensions of Srivastava’s triple hypergeometric functions HA, HB and HC, by using an modified beta function, which given with a generalized M-series in its kernel. We also introduce a new extension of Appell’s hypergeometric function of the first kind by using the same modified beta function. Furthermore, we give some integral representations of the new extensions of Srivastava’s triple hypergeometric functions. Finally, we obtain solutions of fractional differential equations involving new extensions of Srivastava’s triple hypergeometric functions.

Fusing Higher and Lower-Order Biological Information for Drug Repositioning via Graph Representation Learning

Drug repositioning is a promising drug development technique to identify new indications for existing drugs. However, existing computational models only make use of lower-order biological information at the level of individual drugs, diseases and their associations, but few of them can take into account higher-order connectivity patterns presented in biological heterogeneous information networks (HINs). In this work, we propose a novel graph representation learning model, namely FuHLDR, for drug repositioning by fusing higher and lower-order biological information. Specifically, given a HIN, FuHLDR first learns the representations of drugs and diseases at a lower-order level by considering their biological attributes and drug-disease associations (DDAs) through a graph convolutional network model. Then, a meta-path-based strategy is designed to obtain their higher-order representations involving the associations among drugs, proteins and diseases. Their integrated representations are thus determined by fusing higher and lower-order representations, and finally a Random Vector Functional Link Network is employed by FuHLDR to identify novel DDAs. Experimental results on two benchmark datasets demonstrate that FuHLDR performs better than several state-of-the-art drug repositioning models. Furthermore, our case studies on Alzheimer's disease and Breast neoplasms indicate that the rich higher-order biological information gains new insight into drug repositioning with improved accuracy.

The $ m $-weak group inverse for rectangular matrices

<abstract><p>An extension of the $ m $-weak group inverse (or $ m $-WGI) on the set of rectangular matrices is provided to solve some systems of matrix equations. The extension is termed as the $ W $-weighted $ m $-WGI (or $ W $-$ m $-WGI). The $ W $-$ m $-WGI presents a new, wider class of generalized inverses which involves some already defined generalized inverses, such as the $ m $-WGI, $ W $-weighted weak group, and $ W $-weighted Drazin inverse. Basic properties and diverse characterizations are proved for $ W $-$ m $-WGI. Several expressions for computing $ W $-$ m $-WGI are proposed in terms of known generalized inverses and projectors, as well as its limit and integral representations. The $ W $-$ m $-WGI class is utilized to solve some linear matrix equations and express their general solutions. Some new properties of the weighted generalized group inverse and recognized properties of the $ W $-weighted Drazin inverse are obtained as corollaries. Numerical and symbolic test examples are presented to verify the obtained results.</p></abstract>

Некоторые формулы дробного интегрирования от одной функции Фокса с пятью параметрами

The study obtained Riemann – Liouville and Erdelyi – Kober formulas for fractional integration of five-parameter special Fox function. The paper gives an integral representation of the function under consideration in terms of the Mellin – Barnes integral, conditions for absolute convergence, and asymptotic expansions both for large and small values of the argument. We obtained the formulas proved, using the above integral representation and known power rules for integration. For partial parameters, we got some well-known elementary and special functions.

Some Properties of New Hypergeometric Functions in Four Variables

In this paper, we introduce ten new quadruple hypergeometric series. We also obtain their various properties such that integral representations, fractional derivatives, N-fractional connections, operational relations and generating functions.

Drazin theta and theta Drazin matrices

<p style='text-indent:20px;'>In this paper, two new classes of matrices - Drazin - theta matrix and theta - Drazin matrix are introduced for a square matrix of index <inline-formula><tex-math id="M3">\begin{document}$ m $\end{document}</tex-math></inline-formula>. Whenever the index is equal to one, we get special case of matrices called Group - theta matrix and theta - Group matrix respectively. Several characterizations of these matrices, the integral representations, representation in limit form and the representation in terms of rank factorization are obtained. Also, the relationship of Drazin -theta and theta - Drazin matrices with other well known generalized inverses are investigated. By applying the concept of Drazin - theta matrix, general solutions of certain types of matrix equations are characterized here.</p>

Заметка об обобщенной q-гипергеометрической функции типа Райта

In 2001, Virchenko et al. published a paper on a new generalization of Gauss hypergeometric function, namely Wright-type generalized hypergeometric function. Present work aims to define the q-analogue generalized hypergeometric function, which reduces to generalized hypegeometric function by letting q tends to one, and study some new properties. Convergence of the series defining generalized q-hypergeometric function and properties including certain differentiation formulae and integral representations have been deduced.

Revisiting Diffusive Representations for Enhanced Numerical Approximation of Fractional Integrals

This study reexamines diffusive representations for fractional integrals with the goal of pioneering new variants of such representations. These variants aim to offer highly efficient numerical algorithms for the approximate computation of fractional integrals. The approach seamlessly aligns with established techniques used in addressing problems involving integer-order operators, contributing to a unified framework for numerical solutions.

Novel Variants of Diffusive Representation of Fractional Integrals: Construction and Numerical Computation

In this paper, we revisit the diffusive representations of fractional integrals established in Diethelm (2023a) to explore novel variants of such representations which provide highly efficient numerical algorithms for the approximate numerical evaluation of fractional integrals.

Integral Representation of Functions on the Circle

Integral Representation of Functions on the Circle

A recursion relation for the number of Goldbach partitions of an even integer

The contour integral representation of the number of Goldbach partitions of an even integer, G(n), is extended to an integral with a support function that equals a linear combination of integers {G(m)}. A support function is found such that there is a nontrivial integral relation relating number of Goldbach partitions of n and m < n. The proof of the existence of a partition of any even integer greater than or equal to four into the sum of two primes follows from a recursion relation, resulting from an integral identity, that yields a non-zero lower bound for G(n). A partition of n, given a partition of n/2 , is derived from an algorithm based on solutions to congruence relations related to the roots of unity. This result provides a method for generating a prime partition of every even integer greater than or equal to 6 in addition to the above demonstration of its existence.

Laun's rule for predicting the first normal stress coefficient in complex fluids: A comprehensive investigation using fractional calculus

Laun's rule [H. M. Laun, “Prediction of elastic strains of polymer melts in shear and elongation,” J. Rheol. 30, 459–501 (1986).] is commonly used for evaluating the rate-dependent first normal stress coefficient from the frequency dependence of the complex modulus. We investigate the mathematical conditions underlying the validity of Laun's relationship by employing the time-strain–separable Wagner constitutive formulation to develop an integral expression for the first normal stress coefficient of a complex fluid in steady shear flow. We utilize the fractional Maxwell liquid model to describe the linear relaxation dynamics compactly and accurately and incorporate material nonlinearities using a generalized damping function of Soskey–Winter form. We evaluate this integral representation of the first normal stress coefficient numerically and compare the predictions with Laun's empirical expression. For materials with a broad relaxation spectrum and sufficiently strong strain softening, Laun's relationship enables measurements of linear viscoelastic data to predict the general functional form of the first normal stress coefficient but often with a noticeable quantitative offset. Its predictive power can be enhanced by augmenting the original expression with an adjustable power-law index that is based on the linear viscoelastic characteristics of the specific material being considered. We develop an analytical expression enabling the calculation of the optimal power-law index from the frequency dependence of the viscoelastic spectrum and the strain-softening characteristics of the material. To illustrate this new framework, we analyze published data for an entangled polymer melt and for a semiflexible polymer solution; in both cases our new approach shows significantly improved prediction of the experimentally measured first normal stress coefficient.

Faster than FFT: Conformal accelerations method

Fourier-Laplace transform technique allows one to represent several classes of important probability distributions and solutions of basic boundary problems for wide classes of fractional differential equations as integrals of functions enjoying two key properties: analytic continuation to a cone or the union of a cone and tube domain, and regular decay at infinity. Integral representations for Wiener-Hopf factors, fractional moments and special functions enjoy these properties as well. In the paper, we present the general methodology which allows one to evaluate the integrals enjoying these properties very fast and accurately. Among applications, we derive new efficient realizations of the Fourier, Laplace and Z-transforms, representations for probability distributions in Levy models, stable ones including, and algorithms for pricing contingent claims, Monte-Carlo simulations and evaluation of special functions.